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We study cosmological effects of explicit Peccei-Quinn breaking on the QCDaxion dark matter. We find that the axion abundance decreases or increasessignificantly depending on the initial position, even for a tiny Peccei-Quinnbreaking that satisfies the experimental bound of the neutron electric dipolemeasurements. If the axion first starts to oscillate around a wrong vacuum andif it gets trapped there until the false vacuum disappears due tonon-perturbative QCD effects, its abundance increases significantly and isindependent of the decay constant $f_a$, as first pointed out in [JHEP 06(2016) 150]. Thus, the axion produced by the trapping mechanism can explaindark matter even when the decay constant is close to the lower limit due tostellar cooling arguments. On the other hand, if the axion starts to oscillateabout a potential minimum close to the low-energy vacuum, its abundance issignificantly reduced because of the adiabatic suppression mechanism. Thisrelaxes the upper limit of the axion window to large values of $f_a$. We alsodiscuss how the axionic isocurvature perturbation is affected by thePeccei-Quinn breaking term, and show that it can be suppressed in both regimes.In particular, the isocurvature bound on the inflation scale is relaxed by manyorders of magnitudes for $f_a \lesssim 10^{11}{\rm GeV}$ compared to theconventional scenario.

Cosmological effects of Peccei-Quinn symmetry breaking on QCD axion dark matter